Method and Apparatus for Fractal Identification

ABSTRACT

A method and system for applying and reading a fractal image to and from a plurality of objects to act as an identification label is provided. The system includes a printer for printing a fractal pattern to the plurality of objects and a reader for reading the printed fractal pattern. Such a fractal image is robust to printing and imaging difficulties and inconsistencies, and is difficult to copy, thus defending against counterfeiting.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of and incorporates by reference theentire contents of U.S. Provisional Patent Application Ser. No.61/544,056, titled Method and Apparatus for Fractal Identification,filed Oct. 6, 2011 to Hanina; U.S. Provisional Patent Application Ser.No. 61/623,148, titled Method and Apparatus for Fractal-ID™—FractalIdentification and Verification System, filed Apr. 12, 2012 to Hanina etal; U.S. Provisional Patent Application Ser. No. 61/623,193, titledMethod and Apparatus for Fractal-ID™—L-System Fractal CurveIdentification, filed Apr. 12, 2012 to Hanina et al.; and U.S.Provisional Patent Application Ser. No. 61/683,849, titles Method andApparatus for Identification, filed Aug. 16, 2012 to Hanina et al. Thisapplication is also a continuation-in-part of U.S. patent applicationSer. No. 13/388,602, titled Method and Apparatus for FractalIdentification, filed Dec. 28, 2011 to Hanina, the entire contentsthereof being incorporated herein by reference.

TECHNICAL FIELD

This application relates generally to the identification of objectsusing printed identifiers, and more particularly to the identificationof small, uneven, differently shaped, or other objects, such asmedication pills, using one or more identifiers embedded in afractal-based printed identifier (or an identifier based upon anothermathematical system) to create a visually complex fingerprint.

BACKGROUND Counterfeit Medication

A counterfeit medication or a counterfeit drug is a medication orpharmaceutical product which is produced and sold with the intent todeceptively represent its origin, authenticity or effectiveness. Acounterfeit drug may contain inappropriate quantities of activeingredients or none at all, may be improperly processed within the body(e.g., absorption by the body), may contain ingredients that are not onthe label (which may or may not be harmful), or may be supplied withinaccurate or fake packaging and labeling. Counterfeit medicinal drugsinclude those with less or none of the stated active ingredients, withadded, sometimes hazardous, adulterants, substituted ingredients,completely misrepresented, or sold with a false brand name. Otherwiselegitimate drugs that have passed their date of expiry are sometimesremarked with false dates. Low-quality counterfeit medication may causeany of several dangerous health consequences including side effects,allergic reactions, in addition to their obvious lack of efficacy due tohaving less or none of their active ingredients. Medicines which aredeliberately mislabeled in order to deceive consumers—includingmislabeled but otherwise genuine generic drugs—are counterfeit.

Since counterfeiting is difficult to detect, investigate, quantify, orstop, the quantity of counterfeit medication is difficult to determineCounterfeiting occurs throughout the world, although there are claimsthat it is more common in some developing countries with weak regulatoryor enforcement regimes. It is estimated that more than 10% of drugsworldwide are counterfeit, and in some countries more than 50% of thedrug supply is counterfeit. In 2003, the World Health Organizationestimated that the annual earnings of counterfeit drugs were over US$32billion.

The considerable difference between the cost of manufacturingcounterfeit medication and price that counterfeiters charge is alucrative incentive. Fake antibiotics with a low concentration of theactive ingredients can do damage worldwide by stimulating thedevelopment of drug resistance in surviving bacteria. Courses ofantibiotic treatment which are not completed can be dangerous or evenlife threatening. If a low potency counterfeit drug is involved,completion of a course of treatment cannot be fully effective.Counterfeit drugs have even been known to have been involved in clinicaldrug trials.

Medication Identification

In addition to the problem with counterfeit medications, simpleidentification of medication is also an extremely large problem. Morethan 80% of adults in the U.S. take at least one pill a week, whetherprescription, OTC, vitamin or herbal. Yet the pills they are taking aredifficult to identify based on their visual characteristics alone. Pillidentification, or the inability to correctly visually identify a pill,is a large contributing factor to medication errors. These errors canoccur anywhere along the drug-taking process. Difficulty with pillidentification is further exacerbated when patients are older, have someform of impairment, take multiple drugs or have limited health literacy.1.5 million people are harmed each year because of medication errors Thecost of treating drug-related injuries in hospitals is approximately$3.5 billion per year. The actual number of medication errors ispresumably much higher since not all medication errors lead to injury ordeath. A pill's poor labeling and packaging are thought to cause onethird of medication errors, while studies have also shown that a pill'sshape and color are important factors in drug identification.

Existing Identification and Anti-Counterfeiting Technologies

There are several technologies that have been employed in an effort tocombat the counterfeit drug problem, and to allow for identification ofmedication. An example is radio frequency identification which useselectronic devices to track and identify items, such as pharmaceuticalproducts, by assigning individual serial numbers to the containersholding each product. The U.S. Food and Drug Administration (FDA) isworking towards an Electronic pedigree (ePedigree) system to track drugsfrom factory to pharmacy. This technology may prevent the diversion orcounterfeiting of drugs by allowing wholesalers and pharmacists todetermine the identity and dosage of individual products. Sometechniques, such as Raman spectroscopy and Energy Dispersive X-RayDiffraction (EDXR) can be used to discover counterfeit drugs while stillinside their packaging. Other more traditional systems may be applied tosuch medication identification, such as barcoding being provided onmedication packaging (either one or two dimensional). For such a use,however, any damage to the barcode, difficulty in printing the barcode(such as deformation based upon printing surface), or obscuring aportion of the barcode may render the barcode inoperative.

Marking individual pills with one or more identifiers is considered auseful method for identification, but has been traditionally thought ofbeing cost prohibitive while offering only minimal improvement overpackaging marking. One or more barcodes may be employed (either one ortwo dimensional) and may be printed to individual medication pills,instead of, or in addition to being printed to the medication packaging.Such a printing process may be implemented by employing one or moreappropriate printing apparatuses, such as a pad printing apparatusprovided by Printing International® N.V./S.A., for example. Thus, eachpill may be individually printed with the use of such a pad printingapparatus. Laser marking has also been used to print high-resolutionimages or barcodes directly onto pills. In consumable products, Mars®,Inc. utilizes inkjet or pad printers to print images cheaply ontoindividual pieces of candy. Indeed, U.S. Pat. No. 7,311,045 describes asystem for printing multi color images on a candy by maintaining adirectional registration of the candy between printing steps. In eachinstance, holding each individual medication pill or candy is performedby vacuuming the pieces in place, and holding the piece firmly in placebetween steps so that orientation of the piece during printing does notchange. Other patents and applications assigned to Mars®, Inc. describea number of systems and methods for printing food grade inks onto shapedcandy elements.

While one or two dimensional barcodes have been used to serializeindividual pills and verify authenticity and identity, but as recognizedby the inventors of the present invention, their designs are relativelyeasy to replicate, require fixed surface areas and specific alignmentfor printing, and are rendered unusable if occlusion occurs due tohandling or if the barcode is damaged. Unlike forensic features, whichare embedded into an item, in barcode technology the item's physicalattributes are completely distinct from the barcode itself. Further,whether using such a pad printing process, or employing other printingmethods such as ink jet printing or laser marking for imparting markingsto candy or medication pills, the inventors of the present inventionhave recognized that the need for purposefully handling individual pillsmay be time consuming and expensive. Further, the described printedelements may fail to provide robust images sufficient to act as a uniqueidentifier for a particular batch of processed elements. For such a use,as noted, any damage to the barcode, difficulty in printing the barcode(such as deformation based upon printing surface), or obscuring aportion of the barcode may render the barcode inoperative. Additionally,barcodes may be easily copied and applied to counterfeit objects. Noneof these systems are sufficient for imparting robust identificationinformation to a pill or other candy object.

Similar problems of identification of other products or objects, such asconsumer products and the like, may also arise. While holographicprinting on hang tags and the like has been employed in an effort tomark such objects, and to perhaps stop counterfeiting of these objects,these tags may be removed and possibly copied as printing on a singletag may not be seen as a particularly difficult deterrent. Thus, notonly is secure identification impossible, varying levels of desiredsecurity cannot be employed.

Therefore, it would be desirable to provide a method and apparatus thatovercome the drawbacks of the prior art.

SUMMARY

In accordance with one or more embodiments of the present invention, astandardized process for labeling and identifying medication and otherobjects is provided, and in particular may comprise a system and methodfor printing an identification pattern to a plurality of regularly orirregularly shaped and arranged objects. In particular, a fractalpattern is preferably printed as the identification pattern onto aplurality of medication pills or other objects. At a later time,computer vision may be applied to read these applied fractal labels toautomatically confirm identification and authenticity of the pills orother objects, irrespective of orientation, partial occlusion, orpartial damage of the printed fractal image. Finally, different levelsof fractal dimensions (the number of times the fractal pattern isrecursively printed within itself) may be printed and therefore readfrom these fractal images employing different resolution imagingdevices, providing varying levels of security and precision in theidentification process while allowing for ease of identification and areduced usability burden. Thus, imaging devices with lower resolutionimaging capabilities may be able to resolve one or two fractaldimensions and provide a lower, less expensive, consumer oriented levelof authentication, while higher resolution imaging devices may be ableto resolve seven or more fractal dimensions, thus providing additionalsecurity where desired.

Security Approaches

The inventors of the present invention have recognized that four schemeshave typically been used by pharmaceutical manufacturers to identify andauthenticate medication.

-   -   1) Overt on-product marking including holograms, packaging        graphics, etc. Overt features allow the public to see whether a        medication is false or not, which may be useful during the drug        taking process to flag counterfeits, and to allow individuals to        confirm that they are taking the correct medication.    -   2) Covert marking, including invisible ink, embedded images and        watermarks, etc. Covert features are unidentifiable to the        public and are usually places on the medication packaging. Such        covert markings provide a higher level of security than overt        marking as they are more difficult to copy.    -   3) Forensic marking, including chemical, biological and DNA        taggants. Forensic features, which are integrated into the        physical properties of the medication, are available for even        higher security authentication or where scientifically tested        authentication of the item may be required. Of course,        destruction of the product may be required for authentication,        and a change to the manufacturing process of those items is        necessary for implementation.    -   4) Track and trace solutions, including bar codes,        serialization, Radio Frequency Identification/RFID tags, etc.        Track and trace technologies allow for near real-time medication        tracking throughout the supply chain from the manufacturer to        the pharmacy, and to the patient, and have been typically        employed by entities along the supply chain. While end users        have not been traditionally included in this solution, as noted        above, such systems are becoming more readily available for end        users.        While each approach has advantages and disadvantages, employing        components from all four solutions is desirable to effectively        identify and authenticate medications. While one or more of the        above schemes are typically employed on medication packaging,        the inventors of the present invention have recognized that        labeling the individual pills or capsules may provide an even        more robust solution. This allows for both identification and        authentication to occur even if a medication has been separated        from its packaging. Throughout the supply chain, medications        typically change hands—from the manufacturer to the distributor        to the repackager to a secondary distributor and then to the        pharmacy—many times before they end up with the patient.        Therefore, this ability to identify medications down to the        identity of a single pill, may provide substantial additional        benefit.

Fractal Encoding

Fractal geometry is a branch of mathematics that deals with geometricobjects that are too detailed to be described by standard Euclideangeometry [4]. K. Falconer, Fractal Geometry (Wiley, West Sussex,England, 2003). While there is no precise definition of what constitutesa fractal, there are some properties that most fractals share:

self-similarity: parts of each fractal are similar to the whole in somesense (i.e., geometrical, approximate, statistical).

fine structure: zooming in on a portion of the fractal reveals detail nomatter how small the scale may be.

fractal dimension: the “fractal dimension” of the fractal is greaterthan its topological dimension (i.e., 1 for a line, 2 for a square, 3for a cube).

simple definition: though the fractal itself may be quite complex, itsmathematical description is very simple.

A fractal is a rough or fragmented geometric shape that can be splitinto parts, each of which is (at least approximately) a reduced-sizecopy of the whole, a property called self-similarity. A fractal oftenhas the following features:

It has a fine structure at arbitrarily small scales.

It is too irregular to be easily described in traditional Euclideangeometric language.

It is self-similar (at least approximately or stochastically).

It has a Hausdorff dimension which is greater than its topologicaldimension (although this requirement is not met by space-filling curvessuch as the Hilbert curve).

It has a simple and recursive definition.

Because they appear similar at all levels of magnification (at eachprinted dimension), fractals are often considered to be theoreticallyinfinitely complex (in informal terms).

By encoding one or more pieces of identification information into such afractal through the use of one or more predetermined fractal type,color, or other fractal parameter, and printing or otherwise etching theresulting fractal onto a plurality of medication pills or other objects,a robust and secure medication or object identification scheme may beprovided. Varying resolutions and complexities of such fractals may beemployed in order to impart desired levels of security. More complicatedfractals having a greater number of dimensions (and thus allowing animaging device with higher levels of resolution to recognize the fractalpattern more completely at these greater levels of resolution may beemployed for use with objects needing higher levels of security, asthese more complex fractals are more difficult (or impossible) to copy,requiring at least higher quality printers and readers that may not beeasily available. As further recognized by the inventors of the presentinvention, a low-cost tool such as a webcam combined with computervision software may be provided to a patient to properly image andidentify a medication at a lower level of security.

Once applied, the fractal image provides a robust identification systemthat is resilient against identification when partially occluded, orwhen printing is imperfect because of object shape, position, surfacetexture or the like. Thus, one or more predetermined characteristics ofa pill or other object may be employed to be used as part of anidentifier for the object. In such a manner, not only is the fractalimage used to prevent against counterfeiting as it is difficult orreproduce, but further various characteristics of the object, such ascolor, shape, texture, markings and the like, may combine with such afractal image to produce a unique fractal/object characteristiccombination. As the characteristics of the fractal alone are known, thefractal image may act as a calibration tool to determine any influencethe color, for example, of the pill may have on the actual fractalcolor, thus allowing for an accurate determination of pill color. Insuch a manner, these noted characteristics of the object may cause oneor more distortions in the shape, color, or other attribute of theprinted fractal image, these distortions being potentially resolvableand recognizable at differing desired imaging resolutions, providingdiffering levels of security. While a pill may also distort barcodes andother printed images, barcodes include thick lines and may hide anydetails of geometry changes. It is the unique structure of a fractal,and its multi-dimensional, intricate structure, that makes it ideal forperforming this task as less of the pill surface is obscured, allowingfor additional opportunities for measuring such distortion and othersubtle geometric changes in a medication pill or other item. The use ofreference points in such a printed fractal image, and expected distancestherebetween or orientations thereof, may thus allow for precisedetermination of distortions of the image based upon printing techniquesand pill shape.

Furthermore, because of the self replicability of fractal images, theapplication of these unique fractal identifiers may be generated thatmay be applied substantially simultaneously to a large group ofmedication pills or objects without regard to orientation or relativepositioning of the objects during printing. The resulting identificationimages are robust even if portions of the printed images are notproperly printed, or are damaged, obscured or otherwise occluded. Thus,during a preferred printing process, these pills or other objects needonly be maintained in approximately a single layer during the printingprocess. Strict orientation and arrangements of the medication pillsduring printing or subsequent imaging is not required (in that fractalpatterns are recursive patterns that may be viewed at any level ofdetail and give the same information). Additionally, the entire portionof the fractal need not be properly printed on the pill, providingadditional robustness in the printing process. Thus vacuuming of thepills in a particular orientation for printing is not necessary,allowing a relatively disorganized set of pills to be printed at onetime.

Additionally, multiple fractal patterns may be overlaid, thus producinga more complex identification image. These overlaid fractals may beelectronically combined before printing, thus requiring a singleprinting pass, or may use multiple printing passes, thus, printingmultiple fractals at different times on the pills. These overlaidfractal patterns may also be selected to provide different imagingresults, such as a first fractal image allowing for a more accuratemeasurement of shape and color of the medication pill, and anotherfractal image providing various medication information. Further, byproviding more complete coverage of a pill surface, while simultaneouslyperhaps allowing substantial portions of the pill surface to be viewablealong with the fractal image, the geometry of the pill and any unique oridentifiable geometric characteristics may be more accurately determinedthrough a measurement of distortion of the fractal images by the shapeof the pill.

Therefore in accordance with one or more printing mechanism embodimentsof the invention, because any portion of the fractal image is sufficientto provide all of the information, a conveyer mechanism may be providedfor forwarding, in either a continuous or batch processing manner, aplurality of medication pills to a printing area, in addition to the useof the individual pill printing schemes of the prior art. If forwardedin a batch manner, a fractal image is preferably printed on the areacontaining the medication pills in a manner employing, by way of exampleonly, ink jet printing technology of the type described above withreference to the '045 patent, the contents thereof being incorporatedherein by reference. Other printing or etching technologies, such aslaser etching, laser marking, photographic exposure, chemical etching,photolithography, solid ink printing or the like may be employed, eachpreferably providing different and varying possible resultingresolutions for offering differing levels of security. Additionally, oneor more medication pills may be coated with a laser or light sensitivematerial that allows for marking of the pill, while being consumable byhumans. However, unlike the '045 patent, rather than managing objectspiece by piece, in accordance with the present invention, because of thenature of fractal images, a single image may be provided to theplurality of medicine pills at one time. In such a manner, each pillwill receive only a portion of such an image. Because of the selfreferential nature of fractals, this part of the image will besufficient to provide an identifiable amount of information that may berecovered with computer visual recognition technology. Of course,batches of such pills may be manually or otherwise placed in a printingarea, and removed after printing.

If forwarded in a continuous manner, a fractal pattern that may havebounds in the direction across the direction of travel of the pluralityof pills, but is continuously repeatable in the direction of travel, maybe applied. In this manner, a very large number of such pills may beprocessed continuously and inexpensively, and therefore in accordancewith other continuous manufacturing and processing of such pills.

While the invention is described as relating to medication pills, theinvention may be applied to any objects in which such an identificationapplication may be beneficial, including any type of small, uneven, orirregularly-shaped object part, including for pricing, and anywhere atraditional bar code or two dimensional bar code may be employed, andwhere the integrity of the item makes it important to track and identifythe object.

L-Systems

In The Fractal Geometry of Nature (W.J. Freedman and Co., New York,2^(nd) Edition, 1983), Benoit Mandelbrot described a method for theconstruction of a Koch curve using simple geometric figures. Startingwith an initiator made up of connected line segments, at each iterationeach line segment is replaced with a scaled copy of a generator.Repeating this process infinitely many times yields a fractal curve.FIG. 21 illustrates this construction for the first few iterations.Fractal curves can also be represented symbolically using a formalstring-rewriting system known as L-systems. Originally developed byAristid Lindenmayer (Prusinkiewicz, P. & Lindenmayer, A., TheAlgorithmic Beauty of Plants. Springer-Verlag, New York. Retrieved fromhttp://www.algorithmicbotany.org (Original work published 1990).) tostudy the growth of plants, L-systems consist of an alphabet (a set ofcharacters) and production rules, which specify the rules for characterreplacement. Similar to Chomsky grammars, the main difference is thatL-systems apply production rules simultaneously rather thansequentially. If each line segment of an initiator is represented by asingle character and the production rules substitute this character witha string representing a generator, then there is a direct mappingbetween L-system strings and fractal curves. This mapping provides theability to represent a fractal curve in a computer as a sequence ofcharacters.

In order to map strings of characters into geometric objects, turtlegraphics have been developed. The “turtle” marks a position on a plane.It's movement is dictated by the string generated by an L-system. Eachcharacter of the L-system is given a geometric interpretation, i.e.,move forward by distance and draw a line segment, turn right by angle,turn left by angle, etc. FIG. 22, known as the quadratic Koch curve,illustrates this construction using L-systems. FIG. 22( a) illustrates astring representation of the initiator, FIG. 22( b) depicts arepresentation of a generator, while FIGS. 22 (c), (d) and (e) depictiterations of the L-system. The production rule for this generator isF→F+F−F−FF+F+F−F. Here, “F” means moves forward by one unit and draw aline segment, “+” means turn left, “−” means turn right, and is set at90 degrees.

An interesting subset of fractal curves is the set of space fillingcurves. The name derives from the fact that with each iteration of oneof these curves, it fills up an increasing amount of the area of aclosed planar region; taken to the limit, a space filling curve willintersect every point within the region. FIG. 24 illustrates theconstruction of a Peano space filling curve, named after Guiseppe Peanowho first discovered them. As is shown in FIG. 23, the alphabet is theset {F; +; −; R; L} where {F; +; −) have the same interpretation as thequadratic Koch curve discussed above, and {R. L} are place-holdercharacters used to generate the L-system strings. The initiator is L,and there are two productions rules: p₁: L→LFRFL−F−RFLFR+F+LFRFL and p₂:R→RFLFR+F+LFRFL−F−RFLFR.

Current barcoding technology consists of traditional one-dimensionalbarcodes, typified by UPC codes, and newer two-dimensional matrixbarcodes like MaxiCode and QR Code. One-dimensional barcodes encode andprint data in a linear manner using alternating bars of varying width,while two-dimensional barcodes encode data in a matrix format. Bothformats require the barcode to be scanned and placed in properalignment. Both systems rely on a template: decoders extract informationbased on the location of bars, dots, or squares within a well-definedregion. Error-detection methods are typically employed to increase therobustness of the systems, at the cost of decreased information capacityand/or increased barcode size.

Iterated Function Systems

Fractals can be represented as an Iterated function systems (IFS). IFSswere first introduced by Hutchinson (J. E. Hutchinson, “Fractals andself-similarity,” Indiana Univ. Math. J. 30 (1981), pp. 713-747), andpopularized by Barnsley. M. F. Barnsley, Fractals Everywhere (AcademicPress, San Diego, 1988). IFS provide a simple and convenient definitionfor many fractals. An IFS consists of a finite set of contractionsS={S1; S2, : : : , S_(m)}, m≦2, where the Si's are afine maps, i.e.,

(1)

With |det(S_(i))|=|a_(i)d_(i)−b_(i)c_(i)|<1. If the scaling in the x andy directions are equal, the contraction is a similarity.

Given a closed subset D⊂R″ (typically, D=R″), a non-empty subset A⊂D iscalled an attractor of the IFS if A is the fixed point of S, i.e.,

$\begin{matrix}{A = {{S(A)} = {\underset{i = 1}{\bigcup\limits^{m}}{S_{i}(A)}}}} & (2)\end{matrix}$

IFS's have unique attractors, some of which may be fractals. The typicalrequirement for an attractor to be a fractal is that it have non-integerdimension (this is discussed below). If all the contractions in the IFSare similarities, then the fractal is self-similar, i.e., composed ofexact copies of itself. Otherwise, the fractal is self-affine, i.e.,approximately self-similar.

From the definition of an IFS, it is clear that the sets of numericalvalues {a_(i), b, . . . , f_(i)} for i=1, 2, . . . , m completelydetermine the IFS. An automatic generation process may be implemented inthe following manner: for each i=1, 2, . . . , m, 6 numbers arepreferably randomly chosen to form an affine map. The absolute value ofthe determinant of the affine map may then be computed; if it is lessthan 1 it is accepted, otherwise 6 new numbers are preferably randomlyselected. This process may be repeated until m contraction mappings havebeen accepted. The attractor of the IFS may then be computed usingeither the deterministic or random iteration algorithm, and tested forits fractality with user-defined parameters such as fractal dimension.If accepted, the fractal can then be saved in a database by thenumerical values comprising its IFS representation, along with itsfractal parameters.

Two common methods for displaying attractors are 1) the deterministicalgorithm (which follows directly from equation 2); and 2) a randomiteration algorithm. For the deterministic algorithm, starting with aninitial set of points, the contraction mappings are applied in sequence,and the union of the mapped points is computed. This process is iteratedfor a user-defined number of iterations. Good approximations to fractalscan be computed with relatively few iterations. The following algorithmsummarizes the process.

Algorithm 1 Deterministic A₀ ← initial set of points n ← number ofiterations for j = 1 → n do compute S₁(A_(j−1)) ... computeS_(m)(A_(j−1)) A_(j) ← ∪_(i=1) ^(m) S_(i)(A_(j−1)) end for plot A_(n)

Thus, after an initial set of points is defined and a number ofiterations is selected, for each value of j affine maps S1→m arecomputed. Then an output of points based upon all of the computed affinemaps for the particular value of j are combined and then plotted.

FIG. 10 illustrates the deterministic algorithm for the Sierpinskitriangle based upon 10 iterations of the deterministic algorithm for theSierpinski triangle, starting with the unit square. As is shown in FIG.10, an initial set therefore comprises a unit square. A first iterationshows the replacement of each square in the first iteration with a setof three smaller squares. In each subsequent iteration, a similarsubstitution is made, replacing each square in the current iterationwith the three square configuration. As shown, within a small number ofiterations, the complexity of the image is greatly increased. By a shown10^(th) iteration, substantial complexity is achieved. This fractal isgenerated by three contractions:

${S_{1}\left( {x,y} \right)} = {{\begin{pmatrix}0.5 & 0 \\0 & 0.5\end{pmatrix}\begin{pmatrix}x \\y\end{pmatrix}} + \begin{pmatrix}0 \\0\end{pmatrix}}$ ${S_{2}\left( {x,y} \right)} = {{\begin{pmatrix}0.5 & 0 \\0 & 0.5\end{pmatrix}\begin{pmatrix}x \\y\end{pmatrix}} + \begin{pmatrix}0.5 \\0\end{pmatrix}}$ ${S_{3}\left( {x,y} \right)} = {{\begin{pmatrix}0.5 & 0 \\0 & 0.5\end{pmatrix}\begin{pmatrix}x \\y\end{pmatrix}} + \begin{pmatrix}0 \\0.5\end{pmatrix}}$

As mentioned above, attractors of an IFS with non-integer dimension areconsidered fractals (ex: the Sierpinski triangle has dimension D≈1.585).Dimension is a frequently referenced parameter when discussing fractals,and its estimation is an area of much research \cite{Theiler90}.Dimension can be expressed as the exponent that relates the scaling of ageometric object's bulk (area, volume, mass, etc.) with it's size(length, perimeter, diameter, surface area, etc.):

bulk{hacek over ( )}size^(dimension)  (3)

The second method for generating such a fractal, as noted above, maycomprise a random iteration algorithm. Such a random iteration algorithmpreferably works in the following manner according to the followingalgorithm:

select an initial point p₀,

randomly select one of the contraction maps and apply it to p₀ to obtaina new point p₁,

repeat the process for a user-defined number of iterations.

The algorithm preferably produces one point at a time. The individualprobabilities p_(i) may be calculated by:

$p_{i} = {\frac{{\det \; S_{i}}}{\sum\limits_{i = 1}^{m}{{\det \; S_{i}}}} = {\frac{{{a_{i}d_{i}} - {b_{i}c_{i}}}}{\sum\limits_{i = 1}^{m}{{{a_{i}d_{i}} - {b_{i}c_{i}}}}}.}}$

The algorithm preferably outputs a sequence of points which may then beplotted. Detailed images can be created from many (>100,000) iterations.FIG. 11 presents an example of the random iteration algorithm for aself-affine fractal. As the number of points computed increases (movingfrom portions a to b to c in FIG. 11 the algorithm fills in more of theattractor and provides a more accurate approximation. FIG. 11illustrates the random iteration algorithm for the fractal defined bythe following IFS, in accordance with the definitions noted above:

${S_{1}\left( {x,y} \right)} = {{\begin{pmatrix}0.5 & {- 0.4} \\{- 0.6} & {- 0.4}\end{pmatrix}\begin{pmatrix}x \\y\end{pmatrix}} + \begin{pmatrix}{- 0.8} \\{- 0.9}\end{pmatrix}}$ ${S_{2}\left( {x,y} \right)} = {{\begin{pmatrix}0 & {- 0.4} \\{- 1.1} & 0.2\end{pmatrix}\begin{pmatrix}x \\y\end{pmatrix}} + \begin{pmatrix}1 \\0.5\end{pmatrix}}$

L-System Fractal Curve Barcodes

The inventors of the present invention have recognized that one of theprincipal features of fractals is their complexity. Complexity here isdefined as fine detail at arbitrary resolution levels. In the case offractal curves generated by L-systems, a few simple rules may yieldextremely complex curves after just a few iterations. One measure ofcomplexity is the number of line segments created at each iteration. InFIG. 21, the generator is an equilateral triangle, and the initiatorconsists of 4 line segments. Hence, at the th iteration, the curvecontains line segments. At iterations, the Koch curve already has linesegments. In FIGS. 2 and 3, the growth is even greater; at iterations,the quadratic Koch curve has line segments, while the Peanospace-filling curve has line segments per subsquare subsquaresconnecting line segments line segments.

The inventors of the present invention have further determined that thishigh level of complexity provides an opportunity for compactly encodinginformation. If each line segment is drawn with one of two styles(different colors, different line types, etc.), then it can be treatedas a single bit of information. Therefore, in accordance with one ormore embodiments of the present invention, every eight line segments maythen be treated as a byte representing an ASCII character. Since theline segments are drawn sequentially from the string generated by anL-system, order is preserved and can be used to encode strings of data.Using these features, one can therefore encode information directly ontothe fractal curve and treat it as a barcode. To provide robustness toocclusion, data strings can be encoded multiple times on a fractalcurve. Error correction can also be built-in if necessary. In addition,there are many different fractal curves that may be employed, providingvarious quantities of line segments at different sizes and shapes. Also,fractal curve barcodes will be robust to geometric distortion (i.e.,printing on curved surfaces) since it is the sequencing of line segmentsthat matters, not their absolute position on a grid or in a matrix.Alignment will not be an issue, as sequences can be encoded into thefractal curve to signal the beginning and end of the data string.

Kiani et al., described above, have discussed the use of fractal curvesas identifiers. The invention set forth in accordance with the presentapplication differs in several ways from their work:

type of curve;

encoding method;

information extraction.

Kiani et al. only describe the use of Hilbert curves, a type ofspace-filling curve. The Koch curve in FIGS. 22 and 23 are examples ofnon-space-filling curves. The present invention encompasses any fractalcurve that is non-intersecting and can be generated by an L-system.Thus, the reach of the present invention may be applied to a far greatergroup of fractal images.

Kiani et al propose encoding binary information by printing or notprinting line segments. The inventors of the present invention haddetermined that at least one difficulty with this approach is that inthe decoding process, it is impossible to tell whether a line segment ismissing because it represents a particular encoding or if it is missingbecause of occlusion, damage, noise, etc. Therefore, this most importantaspect means that upon acquisition of a sequence of missing linesegments, it is not possible to determine whether these missing segmentsare intended to be missing encoding segments, or whether a portion ofthe image is occluded. Once this inability to differentiate is present,the identification system employing the system cannot be trusted.

The inventors in accordance with the present invention have thereforedetermined that it would be beneficial for all line segments in afractal curve be printed, with multiple printing methods or othermethods for differentiating between two or more types of encoding aparticular line segment, so that the system is more robust to thepreviously mentioned reasons.

Kiani et al use a Hopfield neural network to recover the line segmentsfrom an image of a fractal curve. This requires training a computer torecognize various patterns. Various embodiments of the present inventionemploy line segment detection methods that do not require any training,only standard computer imaging techniques.

Still other objects and advantages of the invention will in part beobvious and will in part be apparent from the specification anddrawings.

The invention accordingly comprises the several steps and the relationof one or more of such steps with respect to each of the others, andthe. apparatus embodying features of construction, combinations ofelements and arrangement of parts that are adapted to affect such steps,all as exemplified in the following detailed disclosure, and the scopeof the invention will be indicated in the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the invention, reference is made tothe following description and accompanying drawings, in which:

FIG. 1 depicts an exemplary medication pill with an exemplary fractalimage printed thereon in accordance with an embodiment of the invention;

FIG. 2 depicts an increased resolution portion of the exemplary fractalimage of FIG. 1;

FIG. 3 depicts an exemplary medication capsule with an exemplary fractalimage printed thereon in accordance with an alternative embodiment ofthe invention;

FIG. 4 depicts a distorted fractal image printed on a portion of amedication pill in accordance with yet another an embodiment of theinvention;

FIG. 5 depicts a conveyer system for conveying batch processing groupsof objects for processing in accordance with an embodiment of theinvention;

FIG. 6 depicts a group of pills for batch processing in accordance withanother embodiment of the invention;

FIG. 7 depicts application of different fractal patterns to a group ofmedication pills in accordance with an embodiment of the invention;

FIG. 8 is a flowchart diagram depicting a printing and imaging processin accordance with an embodiment of the invention;

FIG. 9 is a flowchart diagram depicting image processing in accordancewith an embodiment of the invention;

FIG. 10 illustrates output from a deterministic algorithm for aSierpinski triangle in accordance with an embodiment of the invention;

FIG. 11 depicts an example output from a random iteration algorithm fora self-affine fractal in accordance with an embodiment of the invention;

FIG. 12 illustrates a computation of box counting method for theSierpinski triangle in accordance with an embodiment of the invention;

FIG. 13 depicts application of an IFS to different fractals inaccordance with an embodiment of the invention;

FIG. 14 depicts a deterministic algorithm for the Sierpinski triangle,starting with an initial image including numbers in accordance with anembodiment of the invention;

FIG. 15 depicts a zoomed in portion of FIG. 14;

FIG. 16 depicts encoding of the word “fractal” on a fractal usingcontraction mapping that generated the fractal in accordance with anembodiment of the invention;

FIG. 17 depicts encoding a barcode in different levels of a self-similarfractal-like image in accordance with an embodiment of the invention;

FIG. 18 depicts a fractal image similar to that of FIG. 17, but withadditional complexity added by removing one or more portions of theimage in accordance with an embodiment of the invention;

FIG. 19 depicts a standard fractal progression with modificationsthereto to provide additional complexity to the image in accordance withan embodiment of the invention;

FIG. 20 depicts a plurality of image cells in accordance with anembodiment of the invention;

FIG. 21 depicts a further image including iterations of a complex imagein accordance with an embodiment of the invention;

FIG. 22 depicts initiator, generator and a number of iterations of aKoch curve;

FIG. 23 depicts construction of a quadratic Koch curve using L-systems;

FIG. 24 depicts construction of a Peano space-filling curve usingL-systems;

FIG. 25 depicts encoding of a fractal curve in accordance with anembodiment of the invention; and

FIG. 25 depicts encoding of a fractal curve in accordance with anembodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The detailed embodiments of the invention will now be described makingreference to the following drawings in which like reference numbersdenote like structure or steps.

In accordance with an embodiment of the invention, an ink jet, lasermarking system, or other printing or etching system may be employed inorder to print a plurality of medication pills with a predeterminedfractal image. Other acceptable printing systems may also be employed,and may include, by way of example only, and without limitation, laserprinting, laser etching, photographic imaging, photolithographytechniques, solid ink printing, or the like. One or more known edibleink products may also be employed in the printing process. Invisible, UVsensitive, heat sensitive, and other appropriate inks may be employed.Each medication pill may further be coated with a laser or other lightsensitive or otherwise sensitive coatings that, when caused to react,may be employed to mark the medication pills, while remaining edible byhumans. Furthermore, single or multiple color printing may be employed.

Fractal Printing

Referring first to FIG. 1 depicting a first embodiment of the invention,a fractal image 110 is shown printed to a medication pill 100. As isshown, the fractal image covers a substantial portion of the pillsurface while possible still allowing for viewing of the pill surfacebased upon printing techniques, pill positioning, fractal selection,etc. fractal image 110 may cover substantially less than all of thesurface of medication pill 100.

As is further shown in FIG. 2, a fractal image 110 is once again printedto medication pill 100. Also shown is a magnified portion 210 of fractalimage 110. Because of the self-similarity nature of fractal images, ascan be seen, magnified portion 210 of fractal image 110 lookssubstantially similar to the whole of fractal 110, and is theoreticallyidentical. This self similarity continues, theoretically, to aninfinitely small printed image. Of course in practice, the levels (ordimensions) of self similarity available are limited by printingresolution, and the ability to “see” these multiple dimensions may bedependent upon an imaging resolution of an imaging device. The presentinvention exploits these features of reality in order to provide avaried solution applicable in different security situations.

The printing of fractal patterns is not limited to pill-shapedmedication. As is shown in FIG. 3, such a fractal image 310 may beprinted to a capsule medication 300 comprising first and second capsuleportions 301 and 302. When printed to such a capsule, the fractal imagemay take the shape of the capsule, and may be distorted thereon in apredictable manner. Furthermore, if printed after the capsule has beensealed, any discrepancy between positioning of fractal images on thecapsule portions 301 and 302 may be employed to determine thepossibility of the capsule having been opened or otherwise tamperedwith. Of course, solid capsule shaped pills may also be printed, butwill not include capsule portions.

Fractal Resolution

In accordance with one or more embodiments of the present invention, theuse of high resolution printing and imaging techniques may be employedwhen more robust security measures are required, increasing thedifficulty of copying such an image, and leading to greater accuracy inidentification of each item. Determinations of the maximum resolution ofsuch printed fractals may be identified by building such fractals from asmallest possible printed pattern, thus bounding the lowest fractalresolution. Alternatively, such fractal images may be generated in atraditional form, by starting with a largest fractal image, and thendividing these larger images into smaller and smaller objects, until apredetermined limit or printing resolution is reached. Such increasedresolution may also allow for more precise measurements of the detailsof the fractal image, such as one or more distances between variousportions thereof, ratios of one or more various measured lengths orangles of various portions thereof, etc., texture of the medication pillsurface, color of the medication pill surface, etc. which may then becompared to expected values to confirm authenticity and identification.

Fractal Complexity

Furthermore, various types of fractals may be employed based upon adesired level of security. Thus, more complex versions of fractal imagesmay be employed where a more secure identification system is desired.Variations in such fractal images may include changes in angles, length,number of pixels employed, distribution of one or more characteristicsor pixel density, purposeful omission of particular pixels, use ofparticular color combinations on a planned or randomized basis. Not onlymay generally more complex images be used, but a higher resolutionprinting process may also be employed, thus allowing for more preciseprinting of multiple fractal dimensions, and eventual recall andanalysis of the fractal images by higher resolution imaging devices atdeeper acquisition resolution. Furthermore, as noted above, combinationfractals may be employed to provide additional robustness againstcounterfeiting, and for determining identity. These combined fractalsmay be particularly chosen to allow for determination of different typesof information. Thus, a first or more fractals may be employed tomeasure for detection of distortion that may be a result of the shape ofthe pill, while a second or more fractals may be employed for encodinginformation and for prevention of replication of the fractal images.Additionally, various color gradient application may allow for thecalibration of the fractal image, the pill, or other object.

Referring next to FIG. 8, in accordance with an embodiment of theinvention, a desired level of security may first be defined at step 810.Then, at step 820, a corresponding necessary printing resolution may bedetermined, and at step 830, one or more appropriate printingtechnologies may be identified that will provide sufficient printingresolution to allow for the desired level of security. Finally, at step804, a predefined fractal image is printed to the medication pill usingthe determined printing technology. Of course, if any of steps 810, 820or 830 are predetermined, they can be skipped, or at a minimum,responses to these steps can be predefined.

After printing such a medication, in order to properly identify thepill, imaging steps may be employed. As is further shown in FIG. 8, adesired level of identification security may be defined at step 850, andthereafter, at step 860, a corresponding imaging apparatus may beselected. Thus, if simple visual identification by an end user orpatient is desired, a webcam associated with a mobile device or the likemay be employed to image a high level and one or more additionaldimensions of the fractal image, even if substantially more fractaldimensions have been printed. If, however, full authoritativeanti-counterfeiting identification is desired, an imaging device able toimage to a much higher resolution, thus allowing for the confirmation ofexistence of any desired number of fractal dimensions, may be employed.Next, at step 870, the selected imaging apparatus may be used to acquirean image of the printed fractal and pill. At step 880 the identity ofthe medication pill may be confirmed to the desired level of security.As with printing, if predetermined, any of steps 840 and 850 may beskipped or predetermined (as if the user only has a single imagingdevice).

Thus, as is further shown in FIG. 9, user may present a medication pillwith a fractal identification image printed thereon to a webcam or othermore sophisticated imaging device at step 910. This device may providelocal identification and confirmation of the medication, or may forwardsuch information to a remote location for further processing, and aprocessing step 915 to make this determination may be employed, or sucha determination may be made in advance. If local processing is not to beemployed, then at step 920, an acquired image or video sequence ofimages may be transmitted to a remote location for processing. At step9125, such remote processing may be performed, and at step 930, resultsof such processing may be returned to the imaging device. If at step 915it is determined that local processing is to take place, then processingpasses to step 940 and the pill is analyzed locally. After suchanalysis, the user is notified of the authenticity of the pill at step950. The remote server or local device may analyze the imaged pill,identify the pill, and may indicate a determination of authentication orcounterfeit. If counterfeit (as determined locally or remotely), theuser may be instructed to not take the pill, or alternatively that thepill is authentic in conjunction with step 950.

Fractal Deformation

As is next shown in FIG. 4, a fractal image 410 may be printed to amedication pill 400, and where a portion 411 of fractal image 410 may beprinted on a vertical or other portion 401 of pill 400 other than afront face thereof. In this situation, portion 411 of fractal image 410is printed over a pill feature 412, the edge. Printing over edge 412,and along vertical portion 401 will once again cause the fractal imageto be distorted in a predictable manner. This predictable distortion maybe used to further confirm that the pill is authentic, placing yetanother barrier to a counterfeit medication.

Thus, recognition of predictable distortion of the fractal image andadditional measures to avoid such copying may also be employed inaccordance with various embodiments of the invention. In particular, afractal pattern may be calibrated to include one or more of objectpattern, shape, texture, markings, line thickness (such as through theuse of thicker inkjet lines, or by altering a wavelength of a markinglaser, for example), or the like. As noted above, by including one ormore aspects of the object in the coding scheme and training an imagingsystem to recognize these expected fractal distortions, in order tocounterfeit the object, not only must the counterfeiter precisely copythe fractal image, but must also produce an object nearly identical tothe genuine object with respect to any number of attributes, any ofwhich may pose difficulty. As is further noted above, variouscalibration lines or the use of symmetrical fractal patterns may beemployed for distortion detection. Thus, for example, incorrect objectcolor may change an overall color of the fractal image applied thereto,thus indicating a non-authentic object. If printing resolution isreduced based upon curvature of a pill or other object (and therefore achange in distance from the print head), ink may be distributed in aknown manner, creating a unique signature and allowing for anyrecognition system to better determine the shape of the object moreaccurately. In an additional attempt to provide a difficult to copyimage, as noted above, a plurality of fractal images may be overlaid onan object, thus making copying even more difficult as variousinteractions between the various images may be more difficult todetermine

In addition to including various features of the object in the fractaldefinition, these attributes may influence perception of the fractalimage by an imaging system. Thus, by being printed on a curved surface,for example, a fractal image may be deformed or otherwise influenced inpredictable ways, thus allowing for the user of such shape to beemployed to further differentiate authentic objects. Because orientationof objects is not necessary in accordance with embodiments of thisinvention, deformation of such a fractal pattern may be determined in anumber of likely orientations of an object, and then may be soclassified and found on printed objects. Further, if orientation of theobject can be controlled for printing, then precise fractal deformationmay be determined. As such imaging systems may learn such expectedfractal deformation, the deformation may be employed as part of theobject identification system.

Fractal Coding

In accordance with one or more embodiments of the invention, informationmay be coded into the fractal, by placing such coding information intoone or more parameters that may be stored in a parameter file used togenerate the fractal. Thus, a batch number or the like may be used inthe place of particular parameters to be chosen by the user. Recognitionof the fractal, and reverse engineering thereof to recreate theparameter file may then provide access to the batch number or the likeby the user, for example. It is anticipated that only a predeterminednumber of parameters may be employed for particular coding, others ofthe parameters being adopted to vary in a pseudo-random or otherpredetermined manner in order to make it more difficult to predictfuture likely parameter combinations. Other information that may beencoded in such a fractal may comprise one or more of Medication Name,Dose, Manufacturer, Date of Manufacture, Expiration Date, Location orthe like. More personalized information may also be encoded into thefractal, including Patient Name, Prescription Regimen, Physician Nameand the like. Furthermore, fractal images may be printed at manufacture,at distribution, or in combination. Thus, a manufacturer printed fractalmay be provided for counterfeit prevention, while a second fractal,printed on top of the manufacturer fractal, or alongside thereof, may beprovided and printed by, for example, a pharmacist or the like. Thesefractal images may include particular prescription, patient, prescribingdoctor, date, and other patient and administration specific information.Thus, by allowing for such a combination fractal application, generaland specific identification information may be provided, resulting in arobust, personalized pill marking. Additionally, it may be possible thatone or the other of the manufacturer or local information may beprovided by other than a fractal image. Thus, by way of example, themanufacturer information may be provided in accordance with a fractalinformation, while the particular patient information may be provided bya one or two dimensional barcode, or other information providingprocess. Of course, just the local or manufacturer fractal images may beused.

Alternatively, randomly generated fractals may be employed andrecognized from a look-up table to be associated with a particular batchprocessing unit. Other methods may provide a library of fractal images,and indications of which fractals are to be applied to different type,shaped, or colored objects. Thus, it may be determined that a particulartype of fractal image is best applied to a particularly shaped object.Next, from this subset, a further subset may be determined as best forthe particular color of the object. A fractal image from this subset maythen be used or encoded further, and then applied. As will be furtherdescribed below, such a hierarchical selection process may also speedthe eventual fractal acquisition and recognition process.

Colors may be omitted from the fractal printing process deliberately inorder to increase the number of variations of the code. For example,omission of specific areas may indicate batch number or date.Furthermore, a range of colors may be included with absolute colors suchas black, white, etc. acting as reference points. Use of such a range ofcolors allows for more patterns to be created and utilized, thusincreasing a range of possible unique fractal images. Selection and/oromission of particular pixels in such a fractal image may be furtherused for variation to allow for randomization of predetermined fractalimages.

In accordance with an embodiment of the invention, one may embed codesinto the fractal image that are distributed that can be properlyresolved and interpreted up in reasonable lighting conditions by a lowresolution camera. If higher resolution is available in the fractalimages, but cannot be precisely determined by the low resolution camera,that low resolution camera may further be employed to determine likelydistributions of color or shape across such a fractal that, whileperhaps not being precisely distinctive, do provide an additional levelof security above the simply lower resolution components of the fractalimage. Since these patterns are replicated, the system may decision fusemultiple instances of the same uncertain distribution to come up with amuch higher probability of confirmation.

Printing Techniques

Referring next to FIG. 5, in accordance with an embodiment of theinvention, a conveyer mechanism 510 is shown forwarding one or morebatch processing units 520. Conveyer mechanism 510 is shown as agravity-fed mechanism including a plurality of rollers, but any suchconveyer system may be employed, including gravity-fed, belt driven orotherwise powered conveyer systems, and may further be provided with orwithout a belt system for conveying the batch processing units 520. Ofcourse, any method for forwarding the batch processing elements,including hand delivery of the units, may be employed. Further, conveyermechanism 520 may comprise any desired method, apparatus or system forplacing one or more objects in a location to be imaged in a manner aswill be described below.

It is contemplated in accordance with one or more embodiments of theinvention that each batch processing unit 520 contain a plurality ofindividual objects, and in accordance with a preferred embodiment of theinvention, a plurality of medication pills or the like. Such a pluralityof medication pills 610 are shown in FIG. 6. As shown, medication pills610 are preferably arranged in batch processing unit 520 in anunstructured manner, but generally in a single layer. While slightoverlap may be tolerable in accordance with the invention, a singlelayer presentation of the medication pills will allow for maximumexposure of the pills to an imaging apparatus, shown at 530 in FIG. 1.Such a batch processing unit may comprise from one to any number ofproperly physically locatable pills, and may further comprise a physicalstructure for holding the pills, or may simply comprise a conveyer orother forwarding or holding mechanism for presenting the one or morepills to a printing mechanism. Thus, as batch processing unit 520 isproperly positioned below imaging apparatus 530, imaging apparatus isemployed to administer a predetermined fractal image to the plurality ofmedication pills 610 at one time. Such printing may comprise a rasterprinting system, or may print or otherwise transfer a complete image tothe plurality of medication pills substantially simultaneously. As willbe apparent, each medication pill 610 will be printed with a portion ofthe predetermined fractal image. As noted above, because of the selfreplicating property of fractal images, these portions will includesufficient information to allow for proper identification of variousfractal dimensions at various desired imaging resolutions, thusproviding unique flexibility in imaging based upon desired securitylevels.

Thus, as is shown in FIG. 7, each medication pill 610 is preferablyforwarded for processing to have a fractal pattern imparted thereon.Such a fractal pattern may comprise a repetitive pattern 710, acontinuous pattern 720, or other desired fractal image. Each may be usedin either a batch processing or continuous processing situation. Suchfractal patterns may further comprise one or more combination fractalpatterns, in which two or more fractal patterns are combined to providea resulting complex fractal pattern. These complex patterns may becombined before printing, thus imparting the complex fractal pattern ina single printing pass, or alternatively, each pattern may be printed ina separate pass, thus layering the two or more individual fractals toprovide a resulting complex fractal image.

It may further be desirable to determine where the one or more pills orother objects are located in a single dimension (when printing is inaccordance with a raster type mechanism) or in a two dimensionalarrangement (such as a screen printing system or the like). In such amanner, ink may only be applied where such objects are present, thussaving ink and improving longevity of the system. Furthermore, bydetermining location of objects, and thus potentially batch size,particular fractal images may be employed that are properly suited tosuch batch size or arrangement of objects.

In addition to employing the batch processing method of FIGS. 5 and 6, acontinuous processing system may also be employed. In such a system, asimilar conveyer belt may be employed, in which medication pills orother objects are continuously passed beneath imaging apparatus 530, andmay preferably be employed in conjunction with a continuousmanufacturing process of such objects. A fractal pattern may preferablybe chosen that may be continuously replicated in a direction of travelof the medication pills, while being bounded in the direction acrosstravel, or may be easily repeated in the direction of travel so allmedication pills or other objects are printed with at least a portion ofthe fractal pattern.

Hardware Signature

It has been determined by the inventors of the present invention that aparticular image processing apparatus 530 may affect how the printingink is distributed on the surface of the medication pill or otherobject, thus, in combination with the printed fractal pattern, producinga printer signature, i.e. a printer specific rendering of the particularfractal pattern. The particular characteristics of the printer,including nozzle tolerance, humidity and a host of other factors mayinfluence an output fractal image. Such a printer signature may befurther used as an identifying feature of the printed fractal image.Such a signature may be similarly determined when other imagingtechniques, such as those described above, are employed.

This idea of printer signature may also be extended to product signatureand camera (or more broadly, imaging apparatus) signature. For example,a product signature may be based at least in part on how the productabsorbs the ink, how the pattern at higher resolution distributes theink, texture and reflectivity of the object, shape of the object, etc.Unique texture, shape, color specific to the object (pill) will “code”or distort the fractal into a unique ID. A camera or other imagingapparatus used to eventually image the printed fractal image may alsohave a unique signature in distorting the fractal image that may alsoact as an increase in security as it may be difficult to anticipate acamera that is to be used for imaging, if not an authentic system. Thesesignatures, as opposed to being deficiencies of the system, may beembraced to strengthen the robustness of the system. Such imageinfluences resulting from unrelated characteristics of systems used toimplement the system will be difficult/impossible to replicate. Througha decision fusion process combining the results of analysis of any oneor more of these attributes, an overall picture and confidence ofauthenticity or counterfeit may be determined

Thus, the different hardware and pill interaction signatures, includingshape, texture, color of the object, or the like will all help tofurther “code” or distort the fractal in a unique way. Hence, the uniqueattributes of the product/item will help to create a unique fingerprintfor the fractal. The inventive system is therefore able to learn theunique characteristics of the product through computer vision trainingor the like, and not simply apply an out-of-context code to the item.

At higher resolutions of inkjet printing or other printing, marking oretching processes, codes may be embedded that higher resolution camerasare able to read as well. In the event that inkjet printers can nolonger print at a high enough resolution, then feathering or expectedfeathering distribution may also be picked up based on a distributionthat may be unique to the printer (printer's signature). Alternatively,other higher resolution printing or etching techniques may be employed.In the event that a particular camera, such as a webcam or the like,does not have sufficiently high resolution for acquisition of aparticular fractal image, then expected blurring patterns may be read.This blurring pattern may therefore be provided as a signature initself, and may be learned through computer vision and machine learningor the like by teaching with the lower resolution camera. Multipleinstances of the feathering signature may suggest likelihood ofidentification. Any such recognition system may rely on confidencelevels of detection and confirmation. Thus, even if identification isconfirmed, a threshold may determine confidence over suspicion ofcounterfeit (i.e. how confident the system is that the item isauthentic). Many instances of low confidence levels (even if abovethreshold levels), as received and accumulated over time from any numberof different users at a centralized location, may indicate a potentialcounterfeit issue and raise a flag remotely to anti-counterfeitauthorities to double check a medication source, or alert a user toreport the possibility of a counterfeit medication source.

When printing, and thereafter requesting image acquisition, at higherlevels of resolution, the actual printing method may be employed to codeinformation when continuous lines or images may not be able to beprinted at these higher resolutions. For example, in the case of inkjetdots at very high resolution, the relative positioning of the ink jetdots may be changed, in order to be arranged, for instance, in a form ofa proximity to a center of some printed object or other marker orattribute. Thus, similar to notes on a scale, these same dots may be useto allow for the encoding information even in cases when only lowprinting resolution is available, but high magnification imageacquisition may be available when the pill or object is to beidentified.

Robust Imaging Solution

As noted above, in all cases, because of the inherent replicability offractal images, the solution is effective even if only part of thefractal is printed on the pill. By the nature of fractals, any encodedID is repeated within the shape. Similarly, when reading, if part of thefractal is obscured or otherwise unreadable, the ID may still bedetermined Many varieties of data can be encoded in the fractal (date,time, batch number, location, manufacturer, dose, item etc).Furthermore, writing or reading of the fractal image does not requirealignment of the object perfectly as only part of the fractal may begood enough (may need a percentage of the fractal to be printed on asurface for a specific webcam resolution and inkjet resolution).

Upon reading of a printed fractal identification image through the userof a standard barcode laser scanning device, image acquisition, or othermethod for reading information from the surface of the medication pill,various reference points may be used to aid in determining authenticity.Upon the determination of one or more of such reference pointspreferably defined by one or more of a pattern, color, combinationthereof or the like in a fractal image, distances and ratios to otherreference points may be determined and used to confirm authenticity, ora level of confidence in that authenticity, in a manner as describedabove. Thus, an object may be scanned, and existence of at least aportion of such a fractal may be determined. If there is enoughinformation in the portion of the fractal image to confirm identity,then identity is confirmed. If however, not enough information isavailable, various pieces of fractal images may be pieced together todetermine enough information. Alternatively, an occlusion or the likemay be effectively disregarded through such piecing together of thefractal image. Such information to be pieced together may be taken fromone or multiple fractal dimensions. Further, a user may be instructed tobring such an object closer to an imaging apparatus (or otherwise zoomin on the object), or be asked to switch to another medication pill forre-identification, in order to improve capture resolution. This may beparticularly important in difficult imaging environments, such as in theexistence of bad lighting conditions or the like, which may reduce aconfidence of precision of imaging. Such improved resolution may beemployed alone, or in conjunction with anticipated effects from one ormore object attributes, as noted above, in order to identify anauthentic printed fractal image.

Any applicable imaging system, such as a high resolution system, or awebcam system, will benefit greatly over the use of barcoding. Asorientation of the medication pill is not important, imaging of thefractal image can be performed at any angle of the pill. Furthermore, inaccordance with various embodiments of the invention, various augmentedreality solutions may be employed in order to properly image themedication pill and fractal image, thus truly freeing up the user toimage the fractal image without any real issues regarding orientation orplacement of the pill. Such augmented reality solutions may also provideadditional information regarding the medication pill, including patientname, medication administration schedule information, prescribingdoctor's name, contact information, or any other information that may beuseful for the user to view.

Counterfeit Mapping

Once an authentication, or counterfeit is determined, such informationmay be provided to a remote location to accumulate such information.Each pill identification instance will result in an authenticityconfidence score. With the user's consent, instances of low authenticityconfidence may be reported to a centralized location, along with amedication image, GPS data, as well as time and date stamps.Higher-level authentication tests may be carried out at local pharmaciesusing higher resolution imaging devices. When sufficient notificationsof potential counterfeit medications have accumulated (confidenceflags), a geographic nexus of particular counterfeits will be determinedalongside likely illegal distribution channels, thereby aidinganti-counterfeit officials. Thus, if a high concentration of counterfeititems is found in a location, investigations may be employed in thatarea. Further, proper identifications can be confirmed. Variation inbatch coded information may be employed in order to further allow forconfirmation of particular medication generation time and locationstamps. Such information may be forwarded over the Internet or othertransmission system, such as transmission over a cellular telephoneconnection or the like, to a centralized location for analysis andaccumulation, for example.

Fractal Pattern Selection

Selection of the actual patterns to be employed may be performed inaccordance with consideration of one or more parameters to be encodedinto the fractal image, and further based upon a surface or medicationpill upon which the fractal image is to be printed. As differentinformation may be encoded into each fractal pattern, the selection andencoding of this information will make changes to the fractal pattern insubtle manners. Based upon a printing surface, expected distortion,amount and type of information to be encoded, printing technology to beemployed, or level of resolution in printing and imaging desired,different fractal patterns may be preferred and employed. In fact, eachsuch printed fractal provides a multi-dimensional pattern that comprisesthe above noted fractal signature. These dimensions may include one ormore of fractal image, texture of the surface, color of the medicationpill and shape and contours of the medication pill. These features maybe employed to aid in object recognition.

Furthermore, selection of particular types of base patterns (to bemodified by coding) may be performed in accordance with one or moreparticular tasks, pills or desired results. Thus, for example, one ormore simple fractal patterns, such as a Cantor fractal patterns may beemployed or lower security situations where identification is mostimportant. More complex types of fractal base images may be employed forother, security intense applications.

Image Analysis

For images of objects with a fractal pattern printed on them, computervision techniques may be used to analyze the image. Image analysis mayincorporate such techniques as global and local feature detection anddescription, histogram analysis, image thresholding, image segmentation,etc. Global features may include shape, contours, image moments,object/pattern orientation, etc. Local features may include corners,blobs, line segments, gradients, keypoints, anchor points, etc. Theinformation extracted at this stage may then be used for comparisonbetween test images and stored images. The information may be comparedin various ways, including the use of feature vectors.

Fractal Features

One way fractals are distinguished from regular geometric patterns is bytheir non-integer dimension. For example, the Sierpinski triangle, awell-known fractal, has dimension 1.585. The main definition of fractaldimension is known as Hausdorff dimension, which requires computing aninfinum over all possible coverings of a fractal by small sets. Sincethis covering cannot typically be computed in practice, fractaldimension is usually estimated J. Theiler, “Estimating fractaldimension,” J. Opt. Soc. Am. A 7, 6 (1990), pp. 1055-1073. Commonly usedestimators of dimension are the box-counting dimension and thecorrelation dimension. Both of these are instances of generalizeddimension:

${D_{q} = {\frac{1}{1 - q}{\lim\limits_{r->0}\frac{\log \; {\sum\limits_{i}P_{i}^{q}}}{\log \; r}}}},$

where r is the side-length of the boxes of a grid covering the attractorand P_(i)=μ(

_(i))/μ(

) is the normalized measure of the ith box

_(i). P_(i) is usually estimated by counting the number of points of theattractor in the ith box and dividing by the total number of points.Setting q=0 corresponds to the box-counting dimension while setting q=2corresponds to the correlation dimension. FIG. 12 illustrates thecomputation of the box-counting method for the Sierpinski triangle. Inaccordance with such computation, the attractor is successively coveredwith grids of boxes of decreasing size (as shown in portions a-c of FIG.12. For each such grid, the number of boxes that contain a part of theattractor is counted, and the (box-size˜number of boxes) pairs areplotted on a log-log plot, shown in portion d of FIG. 12. The slope ofthe regression line in this particular instance is calculated as1.58496.

Therefore, in accordance with one or more embodiments of the presentinvention, fractal dimension estimation of an image to be tested may beimplemented in the following manner:

1) Extract fractal from a given test image: The test image may beprovided from any standard camera, such as on cellphones or tablet pc's.Any preprocessing of the image (thresholding, rotation, perspectivetransformations, Gaussian filters, Laplace filters, etc.) may beimplemented here. The image may be applied to any desirable surface, inaccordance with the invention, and may in particular be applied to amedication pill or other object for which authentication oridentification is desired.

2) Apply the box-counting algorithm: The sizes r will typically rangefrom L/2 to 1, where L is the side length of the image.

3) Compare the results of the algorithm to results from stored images:The results calculated in step 2 are preferably compared to results in alist of possible IFS fractals, and the IFS code with the closest fractaldimension is selected. Such a list of possible fractals are preferablyprinted, and this same calculation is performed for each, thusgenerating a table of dimension values for different fractal images thatmay be applied to an object or otherwise printed in accordance with oneor more embodiments of the invention. A stored dimension value that isclosest to the measured and determined fractal value is preferablyindicated as the matching fractal. Additionally, in accordance with oneor more embodiments of the present invention, a user-defined thresholdmay preferably be implemented so that if the difference between thecomputed dimension and expected dimension exceeds the threshold, then noIFS code may be selected from the database and a null value ispreferably returned. The addition of this threshold makes sure that onlyclose images may be considered, and provides a level of confidence thatthere is an actual match between the computed dimension of the imagesfractal image, and the expected dimension of one of the stored fractalimages.

In accordance with an additional embodiment of the invention, aninventive lacunarity analysis may be implemented to determine fractaldimension when the image being analyzed is a fractal. Lacunarity may beestimated by a gliding-box algorithm. C. Allain \& M. Cloitre,“Characterizing the lacunarity of random and deterministic fractalsets,” Physical Review A 44, 6 (1991), pp. 3552-3558. Such a gliding boxalgorithm provides that for each pixel p in an image, a box of radius ris preferably centered at p, and the number of pixels s in the box whichare part of the fractal pattern is counted. This creates a frequencydistribution n(s,r) which is the number of boxes of size r with spixels. After computing n(s,r) for each p and for several box sizes r,the frequency distributions may be converted into probabilitydistributions Q(s,r) by dividing by the total number of boxes N(r) ofsize r. The first and second moments may then be computed as

${Z^{1}(r)} = {\sum\limits_{s}{s\; {Q\left( {s,r} \right)}}}$${{Z^{2}(r)} = {\sum\limits_{s}{s^{2}{Q\left( {s,r} \right)}}}},$

and lacunarity at scale r is then preferably defined as

${\Lambda (r)} = {\frac{Z^{2}(r)}{\left\lbrack {Z^{1}(r)} \right\rbrack^{2}}.}$

For fractal patterns, in accordance with this embodiment of theinvention, plotting log r vs. log A(r) yields a curve which is astraight line with slope m=D−E, where D is the fractal dimension and Eis the Euclidean dimension.

Therefore in accordance with one or more embodiments of the presentinvention, lacunarity estimation of a test image may be implemented inthe following manner:

1) Extract fractal from a given test image: The test image can beprovided from any standard camera, such as on cellphones or tablet pc's.Any preprocessing of the image (thresholding, rotation, perspectivetransformations, etc.) will be implemented here. The image may beapplied to any desirable surface, in accordance with the invention, andmay in particular be applied to a medication pill or other object forwhich authentication or identification is desired.

2) Apply the gliding-box algorithm: The scales r will typically rangefrom 1 to L/2, where L is the side length of the image.

3) Compare the results of the algorithm to results from stored images:The results calculated in step 2 are compared to the results in a listof possible IFS fractals, and the IFS code with the closest lacunarityis selected. Such a list of possible fractals are preferably printed,and this same calculation is performed for each, thus generating a tableof lacunarity values for different fractal images that may be applied toan object or otherwise printed in accordance with one or moreembodiments of the invention. A stored lacunarity value that is closestto the measured and determined lacunarity value is preferably indicatedas the matching fractal. Additionally, in accordance with one or moreembodiments of the present invention, a user-defined threshold maypreferably be implemented so that if the difference between the computedlacunarity and expected lacunarity exceeds the threshold, then no IFScode may be selected from the database and a null value is preferablyreturned. The addition of this threshold makes sure that only closeimages may be considered, and provides a level of confidence that thereis an actual match between the computed lacunarity of the images fractalimage, and the expected lacunarity of one of the stored fractal images.

As lacunarity may be alignment dependent, depending on the fractal imageemployed, a method for aligning such a fractal image may be employed.Thus, in accordance with one or more embodiments of the presentinvention, alignment points may be provided to allow for such alignment.These alignment points may be provided as part of the fractal image(comprising a modification thereof), or may be printed on top of thefractal image to indicate alignment. As alignment is preferred, thislacunarity step may be used alone or in conjunction with the dimensioncalculation described above to determine a matching fractal.

Multi-Fractal Spectrum

To create a feature vector of fractal dimensions, the method describedin the previous section regarding the computation of generalized fractaldimension may be implemented for a user-defined range of values of q,including non-integer values. A Legendre transform may then be appliedto these values β(q),

f(α(q)) = q α(q) − β(q), where${\alpha (q)} = {\frac{{\beta (q)}}{q}.}$

The resulting (α(q), f(α(q))) pairs then form a multi-fractal spectrumfor a user-defined range of α's that may then be used to analyze andcompare images of fractals. K. Falconer, Fractal Geometry (Wiley, WestSussex, England, 2003); Y. Xu, H. Ji, \& C. Fermuller, “A projectiveinvariant for textures,” Computer Vision and Pattern Recognition 2(2006), pp. 1932-1939.

[4, 9].

Evaluation Of a Test Image

Evaluating a test image against a stored image using both image analysisand fractal features requires combining information in a manner thatallows for robust comparison. This process is known as informationfusion (A. Ross \& A. Jain, “Information fusion in biometrics,” PatternRecognition Letters 24 (2003), pp. 2115-2125), and may be implemented atthree levels in accordance with one or more embodiments of theinvention:

1) at the feature extraction level,

2) at the matching score level,

3) at the decision level.

At the feature extraction level, feature vectors generated from imageanalysis and fractal features are preferably combined to form a singlefeature vector to be used in comparison. At the matching score level,image analysis feature vectors from test images may be compared to imageanalysis feature vectors of stored images, fractal feature vectors fromtest images may be compared to fractal feature vectors of stored images,and the scores from the two comparisons are preferably combined. At thedecision level, feature vectors may be classified as accepted orrejected based on user-defined criteria, with a final decision beingmade on the various decisions. Any of the extracted fractal images mayemploy a look up table, locally or remotely online, to determine orotherwise classify the extracted features or otherwise stored images.

In accordance with still further embodiments of the invention, to gainbetter accuracy in evaluation and/or to speed up the process,information selection—using some but not all of the gatheredinformation—may be implemented. Information may be selected usingbootstrapping, boosting, machine learning, or other processes.

Multi-Resolution

As noted in above, fractals exhibit self-similarity at multiple scales.Therefore, in accordance with embodiments of the present invention, thisself-similarity may be leveraged by implementing any of the previouslydescribed algorithms or techniques at multiple resolutions, i.e.,capturing an image at multiple resolutions, filtering an image, resizingan image, etc. The information gathered at each resolution level may becompared to detect the level of self-similarity. This information may beused for both identification and verification purposes. It is thesemultiple levels of resolution capture that may provide additionalrobustness in the image capture process. Using multi-resolution canspeed up the processing. The algorithm may preferably be designed in acascaded manner, including the ability to first check low resolution. Ifthere is a recognition failure (i.e. a wrong identification isconfirmed), processing may stop. If a wrong identification cannot beconfirmed, or if more information is necessary, a higher resolutionlevel may then be considered, and so on. Further, such multi-resolutionis robust to occlusion. Any information missing at a lower resolutionmay be available at a higher resolution because of self-similarity.

By employing such multi resolution systems, situations where preciseidentification and authentication may be required may print such imagesto very small levels of resolution, even requiring a microscope or thelike to read the images. A continuous zooming mechanism may be employedin order to continuously check the fractal pattern at the differentlevels for consistency. Indeed, different imaging devices may beemployed in order to read these images at the different resolutions.Such a reading requirement, and the precision printing required, act toraise the bar for a counterfeit printing process. Merely copying animage may be difficult, and may not result in sufficient printingprecision at the various resolutions. Determining a mathematicalequation used to generate a fractal image may not be possible simply byviewing the fractal image, and thus, printing an image from themathematical equation may be always more precise than copying an image.This differential may provide a method for always ensuring that anauthentic image is able to be resolved a higher zoom levels than acopied image.

Super Resolution

Fractals analysis or high security analysis may require high resolutionimages. However, consumer level cameras have limited capability inresolution, which may result in low resolution images. In accordancewith an embodiment of the present invention, this problem may be solvedby taking advantage of multiple frames using super resolutiontechniques. Super-resolution is a technique to use multiple frames ofthe same object to achieve a higher resolution image. It works only ifthe frames are shifted by fractions of a pixel from each other. Thesuper-resolution algorithm is able to produce a larger image thatcontains the information in the smaller original frames. Therefore, byimaging multiple frames shifted in position slightly from each other,and combining these frames employing super resolution techniques, ahigher resolution image may be obtained for use in accordance with thevarious embodiments of the invention.

Image Quality

A major issue in computer vision is the environment in which imagecapture is taking place, and how any environmental factors affect imagequality. The quality of an image can be influenced by any number ofenvironmental factors. These environmental factors may include lighting,camera resolution, printer quality, capturing methods, camera angle,distance to camera, etc. Any previously described algorithm or techniqueor the information gathered from those algorithms or techniques may beadjusted to account for environmental factors. This adjustment mayinclude filtering images before analysis, adjusting the algorithms ortechniques, weighting of information, etc. Image quality may beestimated both at a global level and a local level.

Once an overall image quality has been determined, it is possible toperform one or more of the following processes on the captured data:

1) reject very low quality samples and guide the user or select the bestsamples from consecutive frames. Or alternatively, recognize that someprinting devices or imaging devices may produce lower resolution images.In these situations, it may be desirable to include special alignment orother points into the fractal images that may be tested to furtherdetermine authenticity. Therefore, in accordance with this particularembodiment of the invention, when an image of low resolution or lowquality is acquired, the special printed points may be employed tofurther detect patterns within the fractal image. Similar to fingerprintanalysis, each pattern can be viewed as a special point. Then theproblem of identification and authentication becomes in part a pointmatching problem. These special points may also be used in identifyingor authenticating higher resolution images. If a high quality image is“zoomed in”, these special points (or smaller patterns) may be detectedwithin each fractal image pattern each pattern. In such a situationsimilar to the lower resolution problem, in addition to the use of thefractal calculations described above, identification and authenticationbecomes in part a larger scale point matching problem, which make thesystem more complex robust and accuracy.

In order to solve such a point matching problem, many approaches may beused in accordance with various embodiments of the invention. Theseapproaches may preferably include one or more of the following:algebraic geometry, including using the information of absolute orrelative positions between two points, absolute distance or relativedistance, i.e count how many points/patterns between two special pointsto reduce the computation burden, the absolute or relative directions ofpoints; Hough transform; relaxation; and energy minimization.

2) isolate regions with low quality and analyze regions with goodquality. Because of the self similarity and repeatability of fractalimages, it is possible to determine portions of an image that provide ahigher resolution of information, and process these portions of theimage. Thus, in accordance with various embodiments of the invention, itis possible to identify one or more desirable portions of an imagerelated to image quality, and process these portions of an image.Furthermore, selective image portion processing may be employed not onlyfor image resolution, but also certain portions may be provided withdifferent colors, allowing for color filters or other color selectiontools to allow for processing of the different portions of the image bydifferent devices, and have these portions isolated and separatelyprocessed.

3) adapt matching strategy based on quality

4) Assign different weights to information (features, matching scoresand etc.) based on quality.

Transformation of Fractals

The attractor equation noted above stated that a fractal is the fixedpoint of an IFS, i.e., it is unchanged when the IFS is applied to it.This process can be observed in FIG. 13. In FIG. 13, portions (a) and(e) display two different fractals, A₁ and A₂. When the S_(i)'s of A₁are applied to both fractals, A₁ remains unchanged, but A₂ is clearlydifferent. Thus, as is shown, portion (b) comprises a singletransformation S₁ applied to A₁, portion (c) comprises a secondtransformation S₂ applied to A₁, and portion (d) comprises the union ofportions (b) and (c). Similarly, portions (f), (g), and (h) comprise theapplication of the same transformations, but to a different fractal. Asis clear from this image, when the various transformations are appliedto the corresponding fractal, the images do not change, other than tochange position. But, when the same various transformations are appliedto a non-corresponding fractal, the fractal images change substantially,in addition to changing position.

In accordance with an embodiment of the invention this process can beimplemented to form an identification system for fractal images based ontransforming them by the contraction mappings of an IFS. A query image qmay be tested against a set R of reference images of fractals bytransforming q by the S_(i)'s of each rεR. The change from q to S(q) maybe measured for each r (any measurement may be user-defined, though theEuclidean norm is the standard method), and the reference image thatresulted in the minimum change may be selected as the closest match. Auser-defined threshold t for the distance measurement may beimplemented, so that if min(d_(i))>t, q is rejected as not being in R.

Information Scaling

The deterministic algorithm for displaying fractals described above,combined with the ability of laser marking technology to print highlydetailed images at extremely small sizes, provides an inventive methodfor printing information at various scales within a fractal inaccordance with various embodiments of the invention. This methodpreferably employs a subset of fractals which satisfy the open-setcondition, i.e., fractals defined by an IFS with contractions S={S₁, S₂,. . . , m≧2, where, given an open set O, the contractions satisfy thefollowing two conditions:

${{\underset{i = 1}{\bigcup\limits^{m}}{S_{i}(O)}} \Subset O},{{{S_{i}(O)}\bigcap{S_{j}(O)}} = {{\varnothing \mspace{14mu} {if}\mspace{14mu} i} \neq {j.}}}$

These two conditions state that fractals which satisfy the open-setcondition can be separated, i.e., the contractions do not overlap. Usingan IFS that meets the open-set condition, an initial image can beprinted into a fractal by repeated iterations of the deterministicalgorithm. This process is illustrated in FIGS. 14 and 15. FIG. 14depicts the deterministic algorithm for the Sierpinski triangle,starting with an initial image including numbers. The algorithm isclearly drawing the same attractor from FIG. 10, while employing thenumbers rather than the unfilled box as in FIG. 10. In particular, FIG.14 shows five iterations after (portions (b)-(f)) after the initialimage (a). FIG. 15 comprises a zoomed in portion of the imaged shown inportion (f) of FIG. 14. As can be seen, the attractor retains thedetails of the starting image, even at a zoomed in, higher resolutionportion thereof. Thus, in accordance with embodiments of the invention,it is desirable to zoom in on one or more portions of a fractal image toretrieve information from this zoomed in portion thereof.

With current laser marking technology able to print a 1080p image intoareas as small as 1 mm², the printed fractal will appear to be a normalfractal, but upon magnification with a microscope, the initial image canbe recovered. A combination of ink jet printing and laser marking mayalso be employed. The self-similarity of fractals that satisfy theopen-set condition ensures that multiple copies of the initial imagewill be clearly visible, providing robustness against occlusion orpartial removal of the fractal. The method may be implemented in thefollowing manner in accordance with an embodiment of the invention:

1) Select an initial image to be scaled down. The image can bearbitrary—it can contain an alphanumeric sequence, an image of aperson's face, corporate logos, etc. The deterministic algorithm ensuresthat regardless of the initial image, the fractal will have the sameshape. This initial image will be repeated a multiple of times in thefinal fractal.

2) Select a fractal that meets the open-set condition. As describedabove, the open-set condition ensures that the contraction maps do notoverlap. The fractal may be chosen arbitrarily or selected by the user.

3) Draw the fractal by the deterministic algorithm. The number ofiterations depends on various parameters, including the size of the areaon which the fractal will be printed, the level of scaling of theinitial image required by the user, and the capabilities of the lasermarking system.

4) Print the fractal onto an object. Printing can be performed via lasermarking or any other method capable of meeting the user's resolutionrequirements.

It is further anticipated that information provided at different scalinglevels of resolution may be imaged sequentially, or in some otherpredetermined manner to confirm consistency of the data through thevarious levels of resolution, thus adding further robustness to thesystem.

Fractal Orbits

In accordance with one or more embodiments of the present invention, thecomplexity of fractals can be utilized in serialization—printing thesame fractal but in different manners, allowing the fractal to carryinformation. This inventive method takes advantage of the fact thatfractals defined by an IFS are the attractors of a discrete timestochastic dynamical system defined by the contraction mappings of theIFS. This can be seen in the random iteration algorithm, where aninitial point is chosen and a sequence of points is generated. In thelanguage of dynamical systems, these sequences are referred to astrajectories, or orbits. Since the fractal is an attractor of thedynamical system, any orbit that starts on the fractal will stay on thefractal. Hence, individual sequences of points can be marked on thefractal, and used for serialization.

In addition to just marking fractal images, in accordance with analternative embodiment of the invention, the orbits themselves can carryinformation. This is most easily seen with IFS's consisting of 2contraction mappings, but can be applied to IFS's with any number ofmappings. With 2 map IFS's, the mappings can be labeled S_(o) and S_(i).An orbit may be generated by function composition, i.e.

a _(k) =S ^(k)(a ₀)=S _(jk)(S_(jk-1)( . . . (S_(j1)),(a₀))),

where j_(k)ε{0, 1}, kε1, 2, . . . . As this equation demonstrates, anorbit can be denoted by the sequence j_(k)j_(k-1) . . . j₁. Since thej_(k)'s are either 0 or 1, there is a 1-1 correspondence between binarynumbers and orbits of an IFS. As a result, sequences of ASCII characterscan be marked on a fractal by the orbit generated by their binaryrepresentation. This is illustrated in FIG. 16, depicting encoding ofthe word “fractal” onto a fractal using the contraction mappings of theIFS that generated the fractal.

This further inventive method can be implemented as follows:

1) Select a fractal to encode. The contraction mappings associated withthe fractal will be used to generate the orbit.

2) Encode an ASCII string containing user-supplied information. Anyprocessing of the ASCII string will be performed in this step: messageencryption, duplication, interleaving, Reed-Solomon error correction,etc.

3) Select a starting point, a₀, for the orbit. This can be designated bythe user or randomly selected.

4) Generate the orbit corresponding to the binary string generated instep 2. The contraction mapping corresponding to the first digit of thebinary string is applied to a₀, and the resulting point a₁=S_(j1)(a₀) isrecorded. Next, generate a₂=S_(j2)(a_(i)). Repeat this process for eachdigit of the binary string.

5) Mark the points of the orbit generated in step 4 on the fractalitself. The points can be marked via special characters, color, or otherconspicuous markings.

Decoding of the sequence can be implemented in accordance with analternative embodiment of the invention as follows:

1) Identify the IFS using any of the previously described techniques.

2) Determine the starting point a₀ of the orbit. This point will bemarked uniquely so it can be easily identified.

3) Determine the remaining points of the orbit that were printed on thefractal. These points may be marked for easy identification.

4) Search for the next point. The contraction maps of the identified IFSare individually applied to a₀, and the resulting points are recorded.

5) Compare the results of step 4. Using a user-defined metric, the orbitpoint closest to a transformed point is selected. The numbercorresponding to the contraction mapping which produced the closesttransformed point is recorded.

6) Loop through the rest of the points. The process described in steps 4and 5 is repeated for each point of the orbit.

7) Decode the orbits. The sequence of digits corresponding to thesequence of contraction mappings that generated the orbit are convertedinto ASCII characters.

Other Fractal Like Images

As noted above, the various embodiments of the invention are alsoapplicable to one or more fractal like images that share manycharacteristics of fractals, but are not necessarily pure fractals inthe mathematical sense. These images may differ from fractals in anynumber of manners. For example, as is first shown in FIG. 17, a versionof a triangle (iteration n=1) including information encoded on each legthereof 1710 a may be provided. After a fractal like iteration(iteration n=2), a similar but smaller triangle including information1710 b is provided. During a pure fractal iteration, each portion 1710 awould be replaced with information 1710 b. However, in accordance withthis embodiment of the present invention, only portions of the new imagethat are presented in areas not already occupied by previous portions ofthe image are presented. Thus, as is shown in the n=2 iteration, twodifferent sizes of information (in this case bar codes) are provided. Asthis progression continues to the n=3 iteration, additional 1710 cinformation is superimposed on the previously provided informationwithout replacing this information. The n=4 iteration further shows1710d information being provided. Any number of desired iterations mayalso be employed. In accordance with this process, information isprovided at different levels of resolution. Whereas the fractal systemprovides overall structure that is replicated at different levels ofresolution, even though the component parts are provided in ever smallerportions, this particular embodiment of the invention provides componentparts of the image at various levels of resolution. In order to provideadditional complexity, the length of each of the individual sides of theimage, and the various angles therebetween may be varied.

As is next shown in FIG. 18, a similar triangular fractal image can beprovided with additional complexity by removing a portion of the initialimage 1810 a (see the missing portion thereof), and by rotating orotherwise changing the position of the initial image when reproduced.Thus, as is shown in FIG. 18, second level portions of the image 1810 bare similar to portions 1810 a, but changed in orientation. A similarchange in orientation is yet again shown by portions of the image 1810c. By this change in position of portions of the image, additionalinformation and robustness may be provided.

As is next shown at FIG. 19, a standard fractal image progression 1010is shown in which each portion of an image is split at each iteration.Thus, as is shown, at iteration n=0, a single line is presented; atiteration n=1, two lines are presented; at iteration n=2, four lines arepresented; at iteration n=3, eight lines are presented; and at iterationn=4, 16 lines are presented. In this manner an exponential progressionof line segments is presented. In accordance with this particularembodiment of the invention, image progression 1020 may be presented,and comprises modification of absolute position and orientation (1030)of one or more of the line segments in each iteration to provide stillfurther complexity in the image. This complexity also comprisesunpredictable information that may aid in providing additionalrobustness to the system.

As is next shown in FIG. 20, a plurality of image cells 1110 may beprovided. These image cells 1110 may comprise any desirable informationtherein. These plurality of image cells may be combined to comprise animage cluster 1120. Thus, the individual image cells may be providedstructure relative to other image cells. Any desirable relationshipbetween these cells may be provided. Additionally, each of these imageclusters 1120 may be further combined to comprise a larger image cluster1130, and may be placed within a single image cell 1110 of a largercell, as desired. This process may be continued to as many levels asdesirable, and thus provides a method of building up an image from asmallest desirable building block, rather than the methods describeabove in which a largest image portion is provided and subdivided toprovide additional complexity to the image.

FIG. 21 depicts a still further embodiment of the invention in which acomplex image 1210 a is provided at a predetermined position and size.Additional predetermined sized versions of this image 1210 b may beprovided at a predetermine positions relative to the initial image 1210a. This information therefore may be provided across an extended area,and provided at predetermined sizes and positions to encode variousinformation. Other additional configurations and sizes may be furtherprovided.

Therefore, in accordance with one or more embodiments of the presentinvention, fractal or fractal like images may be employed to allow forimproved identification and authentication systems and methods. Suchsystems and methods may be applied to any desirable object, and maypreferably be applied to a medication pill to allow for identificationand authentication of medication down to the individual pill level, thusimproving robustness of the identification and authentication systemagainst counterfeit or other identification errors.

L-System Fractals

Encoding

Therefore, in accordance with one or more preferred embodiments of thepresent invention, an inventive encoding process may be implemented inthe following manner, although additional encoding processes may also beemployed:

1. Convert a string of ASCII characters into their 8-bit binaryrepresentation.

2. Apply preprocessing (error correction, message duplication, etc.) tobinary representation.

3. Select fractal curve based on application parameters.

4. Print encoded fractal curve.

Step 1 involves ASCII to binary conversion. As an example, here is thebinary representation of the string, “L-system”:

In this particular example of the embodiment, the string “L-system”contains characters, so the binary representation contains digits. Anyfractal curve (or other mathematical system) with at least line segmentscan be used to encode the string. Of course, other non-binary encodingsystems may be employed, so that each character may comprise one or twoor more states.

Step 2 involves preprocessing of the binary string generated fromstep 1. Error correction methods such as Reed-Solomon can beimplemented, along with redundancy features such as string duplicationor interleaving. This step may also be skipped if desired, in accordancewith various implementations of the invention.

Step 3 involves determining the parameters based on the specificapplication. The size of the preprocessed binary string from step 2 iscompared to parameters including printing area, size of the fractalcurve, and number of line segments needed to encode the binary string.An appropriate fractal curve with the needed number of iterations isselected based on the parameters. Any additional factors may also beemployed in order to determine a desired fractal curve (or othermathematical system), as long as the basic elements set forth herein aremet.

Step 4 involves the printing of the encoded fractal curve based on theinformation gathered from step 3. FIGS. 25 and 26 illustrate thisprocess. FIG. 25 depicts encoding of a fractal curve in which (a) is the2nd iteration of a fractal curve created with a single line segment asthe generator and the initiator from FIG. 2. It contains line segments.(b) is the same curve encoded with the string “L-system”; ′s arerepresented by solid segments, ′s by dashed line segments. FIG. 26 alsodepicts encoding of a fractal curve in which (a) is the 2nd iteration ofthe fractal curve from FIG. 23. It contains line segments. (b) is thesame curve encoded with the string “fractal barcoding technology”; ′sare represented by solid segments, ′s by dashed line segments. Note thesolid region on the curve: this represents a synchronization sequence.

Therefore in accordance with various embodiments of the invention, asthe figures demonstrate, fractal curves can encode a large amount ofinformation in a compact manner.

Decoding

An inventive decoding process for such a generated fractal barcode firstdetermines a type of barcode employed. The determination preferablydepends on whether the barcode is assumed to be generated in accordancewith a template or is accepted as a free form curve embedded in animage. If the decoder expects a fractal barcode in a template, then thedecoder will preferably first extract the fractal barcode from thebackground, rotate and align it, and then check line segments atexpected locations, in effect comparing the portions of the fractalimage to the template. If the fractal barcode is accepted as a free formcurve, then the decoder will preferably determine a starting location onthe curve, determine where the next line segment is located, iteratethrough each sequential line segment, and extract information based onthe characteristics of the line segments. Therefore, in accordance withan embodiment of the invention, the decoding process for a templatedfractal curve can be implemented in the following manner

1. Extract image of fractal barcode.

2. Rotate and scale image.

3. Detect line segments at expected locations and extract binary string.

4. Decode resulting binary string.

Step 1 involves accepting an image of the fractal barcode anddetermining the corner points of the template. As with mosttwo-dimensional barcoding methods, the fractal barcode may be placed ina square region with the corner points clearly delineated for easyrecognition. Of course, and other method for aligning such a fractalbarcode may be employed.

Step 2 involves placing the fractal barcode in proper alignment for linesegment extraction. Since the fractal barcode is preferably extractedfrom real-world images, it may be assumed that the barcode region is notperfectly aligned. Based on the relative location of the corner points,the image is preferably properly rotated and aligned into the desiredregion using a perspective transform, which maps quadrilaterals toquadrilaterals. Other methods of such additional alignment may also beemployed.

Step 3 involves iterating through the sequence of line segments at theirexpected locations. Line segment detection may be performed using theHough transform or other appropriate method, and binary digits arepreferably assigned on whether a line segment has been detected or notat appropriate locations. The output of iterating through all the linesegment locations is a binary string.

Step 4 involves the decoding of the binary string extracted from step 3.

In accordance with an alternative embodiment of the invention, thedecoding process for a free form fractal curve can be implemented in thefollowing manner:

1. Extract image of fractal barcode.

2. Detect line segments.

3. Check status of line segments and extract binary string.

4. Decode resulting binary string.

Step 1 involves accepting an image of the fractal barcode. Norequirements are preferably made on positioning, rotation, alignment,etc. The fractal curve is extracted from the background using standardcomputer vision techniques (thresholding, Canny edge detection, etc.).

Step 2 involves using the Hough transform (or other appropriatemathematical system or the like) to detect line segments. The linesegments are ordered sequentially, so the decoder will preferablyiterate through the fractal curve, extracting the location of each linestring one after another.

Step 3 involves checking the status of each detected line segment. Here,status is preferably the line color or type, but any type of linesegment differentiator may be employed. A binary digit will be assignedto each line segment based on the line type, creating a binary string.

Step 4 involves the decoding of the binary string extracted from step 3.If the fractal curve is closed (as in FIG. 5), then the binary stringwill be synchronized to recover the proper starting position and theresulting binary string decoded. Thus, after traversing the entirefractal curve, the position of the fractal curve may be shifted until astarting point of the fractal curve is properly aligned, thus allowingfor extraction of proper codes. This may include sequentially applyingone or more rules to such a determined binary string to furtherdetermine if such a sequence follows one or more rules for decoding.Other methods may be employed for extracting the information from thefractal curve. As noted above, while a binary encoding system isdescribed, encoding of the one or more line segments may include anynumber of differentiable formats. Thus encoding a system employing anEnglish alphabet may be realized using 26 different types of linesegments in the image. Any other encoding system may also be embodied.

Information Scaling

With the scaling of line segments that occurs in the generation ofL-systems fractals, information can be encoded into each individual linesegment, rather than being spread out over the entire fractal. Withcurrent laser marking technology able to print 1080p images into areasas small as, information can be effectively scaled within a fractalpattern, with retrieval of the pattern on a line segment requiringscanning with a microscope. This results in a secure method forembedding information into a fractal pattern, while still maintainingthe overall characteristics of the fractal. This method can beimplemented in the following manner:

1. Initialize line segment.

2. Select L-system fractal.

3. Determine parameters for printing fractal.

4. Print fractal pattern.

Step 1 comprises initialization of the line segment where informationwill be stored. The user can select any format they choose for encodingthe line segment.

Step 2 comprises selection of an L-system fractal. The fractal can bechosen arbitrarily or selected by the user.

Step 3 comprises parameter determination. Parameters include size ofarea where the fractal will be printed, number of iterations forgeneration of the fractal, limitations of printing technology that willbe used in step 4, and limitations of scanning equipment for informationretrieval.

Step 4 comprises the printing of the resulting fractal pattern from theparameters derived in step 3.

Example Applications:

One or more possible applications are outlined below. This list shouldbe considered exemplary, and should not be construed as limiting theapplication of the inventive technology to other applications.

Such a fractal image may be applied as a security labeling system to anyitem that is created in batches and may be varied in shape, such asmedication pills in the manner as describe above. When combined with afacial recognition system, matching of patient and medication can beperformed. The fractal image may be applied as a game on candy as areplacement to a “scratch and win” system, thus creating a show and winapplication, requiring the showing of the candy with the fractal imagethereon being shown to an imaging device such as a web cam on acomputer, mobile device or the like.

Such a fractal image may be applied to handbags or other fabric/clothingon the inside of a garment or label (as difficult to replicate directlyonto a 3d texture/surface).

Such a fractal image system may also be employed with an identificationsystems employing facial recognition, or other biometric identificationsystem. Thus, identification of a patient or other user may be madeemploying one or more known identification systems, such as those notedabove or others. Pill identification may them be performed, and aconfirmation that the particular identified user is to take theidentified pill. If customized fractal images are to be used, the systemmay be able to determine whether the particular pill being imaged isbeing taken at the right time by the correct person. Thus, through theability to personalize such medication pills by batch number, patient,or particular pill dosage, a link between the pill and user may beestablished and confirmed. Release of such personalized information maytherefore be predicated on proper biometric or other identification.

The following features may also be provided in accordance with theinventive system, as related to reporting of various results ofidentification determinations. Use of the inventive fractalidentification systems may be employed to provide an audit trail for apill or object manufacturer. Thus, upon use of the inventive fractalrecognition system by a consumer, seller, or other individual, it may bepossible to log results and alert the manufacturer if imaged fractalidentifiers show low confidence (based on fractal integration withobject—shape, color, texture, curvature), thus perhaps indicating anintent to replicate or otherwise provide a counterfeit product. Suchinformation, along with location data, may be provided to authorities orother systems for tracking such counterfeiting, and in order todetermine or identify counterfeit drug distribution points, or otherareas with such high counterfeit drugs. Consumers may be provided withan incentive to check the identity of such fractal images, thusincreasing availability of such widely spread identificationinformation.

Similarly, to the extent that imaging of the fractal images determinesthat exact matches are present, similar information indicating positiveresults, and a likely absence of such counterfeiting may be provided tothe manufacturer. Such information may also include coded information,such as batch, time/date, location and other information that may beavailable. Consumers may also be provided this positive matchinformation so that they can be sure that their pill or other item isgenuine, and that their security is being safeguarded.

System Benefits

Benefits of employing the inventive fractal imaging system are myriad.The use of an ink jet printing process is easily available, andrelatively inexpensive while remaining flexible. Other printing oretching processes, such as one or more of those noted above, may beemployed when other combinations of cost and security are to beconsidered, or when mass printing is to be employed. Mass batch andcontinuous processing avoids costs associated with properly printingimages on individual pills or other objects. Thus, the fractal imagesmay be quickly modified, by including changes to the parameters forgenerating the fractal images, thus being indicative of various codingincluded in the fractal image.

A fractal library database that changes over time may be employed, inthe manner noted above, so that changes over time may be documented andlater confirmed. Such a fractal database may also be tailored to webcamresolutions commonly used in smartphones, thus providing fractal imageswith resolution acceptable and able to be imaged by standard webcams insmartphones. Since the ID can be confirmed via a consumer with a mobiledevice and smart phone, it means that no special scanning hardware isneeded. Thus, manufacturers or others may educate consumers to image andverify authenticity of an image, such as through imaging using a webcamon a mobile device or the like.

Confidence levels will allow for different security options. Thusdifferent levels of resolution may be employed based upon a level ofdesired security. Higher resolution images, requiring higher resolutionimaging devices may be employed for higher security applications, whilelower resolution imaging and printing may be employed for lower securityoptions. If the inkjet is a low quality printer, then distribution athigher resolutions (as measured by the camera) may suggest that fraudwas committed if the ink distribution (or one or more other printingattributes) is different to what is expected.

As noted above, since inkjet or other etching or printing technologiesmay allow for rapid changing of patterns, one can quickly update afractal pattern to be printed, and link the pattern to specific dates ofproduction and/or batch numbers. This may be especially useful forperishables or medications that go out of date. Also may be useful forfashion items where dates of production are important. Use of olderfractal patterns on newer objects may suggest counterfeiting, forexample. Inks that fade over time may be employed, thus indicatingpassage of substantial time, or the like. Alternatively, inks that maybe wiped off or otherwise removed may be employed to allow formaintenance in the integrity of the item, while still providing desiredlevels of security.

As described above, it is easy to print whole or part of the fractalimage onto the medication as alignment is not critical. The inventivetechnology allows pills and objects of different shapes, curvature, andsizes to be labeled uniquely. In fact the unique shape, color andtexture of the item allows for a unique ID to be printed and can help todifferentiate close but not identical items. The shape of the pill oritem may have an influence on the way the fractal is printed allowingfor expected distortions in the pattern to be recognized. This robustsystem may be particularly useful in identifying counterfeit items thatmay have slightly different shapes.

As further described above, one or more fractal patterns may have anumber of different colors built in that act as a calibration code. Whenpills or other objects appear to be slightly different in color whencompared to the fractal—pill combination (a known color gradient andrange of the fractal and ratio with the pill), then a warning sign canbe issued. Thus, comparison of the color of the pill or other object tothe fractal, or consideration of an effect the color of the pill orother object may have on the color of the fractal may aid in thedetermination of status of the pill or other object. Furthermore, aseach object surface will have a unique texture, the fractal may bedistorted in a predictable, measurable manner in accordance with thisknown object surface texture. This allows for texture to be identifiedas well of a specific surface and differences flagged.

Fractal identification labeling is far superior to existing imaginglabels because it does not require the whole fractal necessarily to beprinted on the medication or other object. Use of the inventive printedfractal image will allow for occlusion by the user (due to fingersblocking image or poor environmental conditions) as only a portion ofthe image may be required to reach desired confidence levels. Such afractal identification label does not suffer from occlusion problems(e.g. traditionally, one number hidden or obscured may inhibit use ofthe ID system). Computer vision may therefore be used to learn and toidentify replicating blurred images in the fractal and/or expectedratios of patterns, colors or shapes. In the event that occlusion occursdue to the environment and/or finger occlusion, then the system may beable to “piece together” different parts of available fractalimages/sections, to create a complete code within a certainty range.Furthermore, fractal identification may operate better than a number, ormore traditional barcode, as it is more difficult to replicate.

This is the first universal pill labeling system developed throughgeometry. The problem with present day technology is that it does notlabel the pill, or if it does label the pill the information isinaccessible to the public or can be easily copied or destroyed. Thepresent invention is novel because it concentrates solely on theprinting of the pill and the organic nature of the fractal patternsthemselves, and the complexity embedded within them, to make the systemextremely difficult to replicate. The fractal patterns may blanket theentire surface of the pill. Any random segment of the pill, no matterhow small or how large, will be able to be used to identify it. This isa tremendous improvement over barcodes, which are static fixed-formlabels that do not lend themselves to different resolutions, and arealso easily copied, damaged, and do not adapt to the physicalconfiguration of a pill. Progressively higher levels of fractalresolution will also allow for progressively higher levels of securityauthentication. It is the first solution that addresses high securitypill identification needs with public accessibility.

It will thus be seen that the objects set forth above, among those madeapparent from the preceding description, are efficiently attained and,because certain changes may be made in carrying out the above method andin the construction(s) set forth without departing from the spirit andscope of the invention, it is intended that all matter contained in theabove description and shown in the accompanying drawings shall beinterpreted as illustrative and not in a limiting sense.

It is also to be understood that this description is intended to coverall of the generic and specific features of the invention hereindescribed and all statements of the scope of the invention which, as amatter of language, might be said to fall there between.

What is claimed:
 1. A system for providing an identifier to a one or more objects, comprising: a conveyance mechanism for positioning the one or more of objects in a predetermined position; an encoder for defining an encoded fractal pattern; and a printer for printing the encoded fractal pattern to the one or more objects.
 2. The system of claim 1, wherein the one or more objects comprise medication pills.
 3. The system of claim 1, wherein the fractal pattern includes one or more piece of coded information contained therein.
 4. The system of claim 1, wherein a portion of the fractal image is printed on each of the one or more objects.
 5. The system of claim 1, wherein the encoded fractal image is defined in accordance with an Integrated Function System.
 6. The system of claim 1, wherein the encoded fractal image is defined in accordance with an L-system.
 8. A system for reading an identifier from an object, comprising: an imaging apparatus for imaging a portion of a fractal image from the object; a processor for determining one or more attributes of the fractal image; and a comparator for comparing one or more expected attributes of the imaged fractal image with one or more determined corresponding attributes of the imaged fractal image.
 9. The system of claim 8, wherein one of the one or more attributes is determined by box counting.
 10. The system of claim 8, wherein one of the one or more attributes is determined by a process comprising the steps of: rotating and scaling the fractal image; extracting a binary string from the rotated and scaled fractal image; and decoding the binary string.
 11. The system of claim 8, wherein a high resolution imaging apparatus is employed to image a high resolution version of the fractal image so that a predetermined number of iterations of the fractal image are visible.
 12. The system of claim 11, wherein occlusion of a portion of the fractal image results in the use of a smaller iteration of the fractal image, thereby recovering all information therefrom.
 13. A method for providing an identifier to one or more objects, comprising the steps of: providing the one or more objects to a printing location; defining an encoded fractal pattern; and printing a fractal image to the one or more objects.
 14. The method of claim 13, wherein the one or more objects comprise medication pills.
 15. The method of claim 13, wherein the fractal pattern includes one or more piece of coded information contained therein.
 16. The system of claim 13, wherein a portion of the fractal image is printed on each of the one or more objects.
 17. The system of claim 13, wherein the encoded fractal image is defined in accordance with an Integrated Function System.
 18. The system of claim 13, wherein the encoded fractal image is defined in accordance with an L-system.
 19. The method of claim 13, wherein the fractal image is printed with a higher resolution when a higher level of security is desired.
 20. The method of claim 13, wherein the fractal image further comprises a combination of a plurality of fractal images. 